Spectral analysis of dissipative Dirac operators with general boundary conditions. (Q1404916)

The author of this paper deals with a \(2 \times 2\) symmetric system of the differential operators of Dirac type \( A(x)^{-1}[J \frac{d}{dx} + B(x)] \) in \(\mathbb R\), where \(J= \left( \begin{smallmatrix} 0& -1 \\ 1& 0 \end{smallmatrix} \right)\), and \(A(x) >0\) and \(B(x)\) are \(2 \times 2\) Hermitian matrices with elements being real-valued, locally integrable functions in \(\mathbb R\). It is considered as an operator in the Hilbert space \(L_A(\mathbb R; \mathbb C^2)\) of the \(\mathbb C^2\)-valued measurable functions in \(\mathbb R\) with the inner product \((y,z) = \int_{\mathbb R}(A(x) y(x), z(x))_{\mathbb C^2}\, dx\). The main purpose of the present paper is, for the corresponding minimal operator with deficiency indices \((2,2)\), to investigate all its maximal dissipative, selfadjoint, and/or other extensions in terms of boundary conditions at infinity. The dilation theory is also employed.